ON LARGE SUBSETS OF Fnq WITH NO THREE-TERM ARITHMETIC PROGRESSION
arXiv:1605.09223v1 [math.CO] 30 May 2016
JORDAN S. ELLENBERG AND DION GIJSWIJT Abstract. In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of Fnq with no three terms in arithmetic progression by cn with c < q. For q = 3, the problem of finding the largest subset of Fn3 with no three terms in arithmetic progression is called the cap problem. Previously the best known upper bound for the affine cap problem, due to Bateman and Katz [BK12], was on order n−1−ǫ 3n .
The problem of finding large subsets of an abelian group G with no three-term arithmetic progression, or of finding upper bounds for the size of such a subset, has a long history in number theory. The most intense attention has centered on the cases where G is a cyclic group Z/NZ or a vector space (Z/3Z)n , which are in some sense the extreme situations. We denote by r3 (G) the maximal size of a subset of G with no three-term arithmetic progression. The fact that r3 ((Z/3Z)n ) is o(3n ) was first proved by Brown and Buhler [BB82], which was improved to O(3n /n) by Meshulam [Mes95]. The best known upper bound, O(3n /n1+ǫ ), is due to Bateman and Katz [BK12]. The best lower bound, by contrast, is around 2.2n [Ede04]. The problem of arithmetic progressions in (Z/3Z)n has sometimes been seen as a model for the corresponding problem in Z/NZ. We know (for instance, by a construction of Behrend [Beh46]) that r3 (Z/NZ) grows more quickly than N 1−ǫ for every ǫ > 0. Thus it is natural to ask whether r3 ((Z/3Z)n ) grows more quickly than (3 − ǫ)n for every ǫ > 0. In general, there has been no consensus on what the answer to this question should be. In the present paper we settle the question, proving that for all odd primes p, r3 ((Z/pZ)n )1/n is bounded away from p as n grows. The main tool used here is the polynomial method, in particular the use of the polynomial method developed in the breakthrough paper of Croot, Lev, and Pach [CLP16], which drastically improved the best known upper bounds for r3 ((Z/4Z)n ). In this case, they show that a subset of G with no three-term arithmetic progression has size at most cn for some c < 4. In the present paper, we show that the ideas of their paper can be extended to vector spaces over a general finite field. Remark 1. The ideas of this paper were developed independently and essentially simultaneously by the two authors. Since the arguments of our two papers were essentially identical, we present them as joint work. We begin with a slight generalization of Lemma 1 of [CLP16]. Let Fq be a finite field and let n be a positive integer. Let Mn be the set of monomials in x1 , . . . , xn whose degree in each variable is at most q − 1, and let Sn be the Fq -vector space they span. Date: 27 May 2016. The first author is supported by NSF Grant DMS-1402620 and a Guggenheim Fellowship. We thank Terry Tao, Tim Gowers, and Seva Lev for useful discussions during the production of this paper. 1
Fn
Observe that the evaluation map e : Sn → Fq q given by e(p) := (p(a))a∈Fnq is a linear isomorphism. Indeed, both spaces have dimension q n and the map e is surjective since for Q n every a ∈ Fnq the polynomial i=1 (1 − (xi − ai )q−1 ) is mapped to the indicator function of point a. For any real number d in [0, 2n], let Mnd be the set of monomials in Mn of degree at most d and Snd the subspace of Sn they span. Write md for the dimension of Snd . By a slight abuse of notation, we use “polynomial of degree at most d” to mean an element of Snd . Proposition 2. Let Fq be a finite field and let A be a subset of Fnq . Let α, β, γ be three elements of Fq which sum to 0. Suppose P ∈ Snd satisfies P (αa + βb) = 0 for every pair a, b of distinct elements of A. Then the number of a ∈ A for which P (−γa) 6= 0 is at most 2md/2 .
Remark 3. The proof of Proposition 2 is essentially the same as that of Lemma 1 of CrootLev-Pach [CLP16], which proves the proposition in the case (α, β, γ) = (1, −1, 0). In the γ = 0 case, the conclusion of the proposition is that P (0) = 0 once n > 2md/2 ; it turns out to be essential for the present application to have the added flexibility of forcing P to take vanish at a larger set of places. Proof. Any P ∈ Snd is a linear combination of monomials of degree at most d, so we can write X (1) P (αx + βy) = cm,m′ m(x)m′ (y). m,m′ ∈Mnd : deg(mm′ )≤d
In each summand of (1), at least one of m and m′ has degree at most d/2. We can therefore write (not necessarily uniquely) X X P (αx + βy) = m(x)Fm (y) + m(y)Gm (x) d/2
d/2
m∈Mn
m∈Mn
d/2
for some families of polynomials Fm , Gm indexed by m ∈ Mn . Now let B be the A × A matrix whose a, b entry is P (αa + βb). Then X X Bab = m(a)Fm (b) + Gm (a)m(b) d/2
d/2
m∈Mn
m∈Mn
This is an expression of B as a sum of 2md/2 matrices, each one of which visibly has rank 1. Thus the rank of B is at most 2md/2 . On the other hand, our hypothesis on P forces B to be a diagonal matrix. The bound on the rank of B now implies that at most 2md/2 of the diagonal entries of B are nonzero. This completes the proof. Theorem 4. Let α, β, γ be elements of Fq such that α + β + γ = 0 and γ 6= 0, and let A be a subset of Fnq such that the equation αa1 + βa2 + γa3 = 0 has no solutions (a1 , a2 , a3 ) ∈ A3 apart from those with a1 = a2 = a3 . As above, let md be the number of monomials in x1 , . . . , xn with total degree at most d and in which each variable appears with degree at most q − 1. Then |A| ≤ 3m(q−1)n/3 . 2
Proof. Let d be an integer in [0, (q − 1)n]. The space V of polynomials in Snd vanishing on the complement of −γA has dimension at least md − q n + |A|. Write S(A) for the set of all elements of Fq of the form αa1 + βa2 , with a1 and a2 distinct elements of A. Then S(A) is disjoint from −γA by hypothesis, so any P vanishing on the complement of −γA vanishes on S(A). By Proposition 2, we know that P (−γa) is nonzero for at most 2md/2 points a of A, for every P in V . View the elements of V as functions on Fnq and let P ∈ V have maximal support. Let Σ := {a ∈ Fnq : P (a) 6= 0} be the support of P . We have |Σ| ≥ dim V for otherwise, there would exist a nonzero Q ∈ V vanishing on Σ. But then the support of P + Q would strictly contain Σ, contradicting the choice of P . Since the support of P is contained in −γA, we have |Σ| ≤ 2md/2 by Proposition 2, and hence dim V ≤ 2md/2 . It follows that whence
md − q n + |A| ≤ 2md/2 |A| ≤ 2md/2 + (q n − md ).
We note that q n − md is the number of q-power-free monomials whose degree is greater than d; these are naturally in bijection with those monomials whose degree is less than (q − 1)n − d, of which there are at most m(q−1)n−d . Taking d = 2(q − 1)n/3, we thus have |A| ≤ 2m(q−1)n/3 + (q n − m2(q−1)n/3 ) ≤ 3m(q−1)n/3
as claimed.
It is not hard to check that m(q−1)n/3 /q n is exponentially small as n grows with q fixed. We can be more precise. Let X be a variable which takes values 0, 1, . . . , q−1 with probability 1/q each; then m(q−1)n/3 /q n is the probability that n independent copies of X have mean at most (q − 1)/3. This is an example of a large deviation problem. By Cram´er’s theorem [RAS15, §2.4], we have 1 lim log(m(q−1)n/3 /q n ) = −I((q − 1)/3) n→∞ n where I is the rate function of the random variable X, calculated as follows: I(x) is the supremum, over all θ in R, of (2)
θx − log((1 + eθ + . . . + e(q−1)θ )/q).
We note that (2) takes the value 0 at θ = 0 and has nonzero derivative at θ = 0 unless x = (q − 1)/2, so the supremum of (2) is positive; this shows m(q−1)n/3 = O(cn ) for some c < q. Corollary 5. Let A be a subset of (Z/3Z)n containing no three-term arithmetic progression. Then |A| = o(2.756n ). √ Proof. Taking q = 3 and x = 2/3, the supremum in (2) is attained when eθ = ( 33 − 1)/8 and we obtain the bound 3e−I(2/3) < 2.756. The theorem now follows by applying Theorem 4 with α = β = γ = 1. 3
References [BK12] M. Bateman and N. Katz, New bounds on cap sets, Journal of the American Mathematical Society 25 (2012), no. 2, 585–613. [Beh46] F. A Behrend, On sets of integers which contain no three terms in arithmetical progression, Proceedings of the National Academy of Sciences of the United States of America 32 (1946), no. 12, 331. [BB82] T. C. Brown and J. P. Buhler, A density version of a geometric Ramsey theorem, Journal of Combinatorial Theory, Series A 32 (1982), no. 1, 20–34. [CLP16] E. Croot, V. Lev, and P. P. Pach, Progression-free sets in Zn4 are exponentially small (2016). arXiv preprint 1605.01506. [Ede04] Y. Edel, Extensions of generalized product caps, Designs, Codes and Cryptography 31 (2004), no. 1, 5–14. [Mes95] R. Meshulam, On subsets of finite abelian groups with no 3-term arithmetic progressions, Journal of Combinatorial Theory, Series A 71 (1995), no. 1, 168–172. [RAS15] F. Rassoul-Agha and T. Sepp¨ al¨ ainen, A course on large deviations with an introduction to Gibbs measures, Vol. 162, American Mathematical Soc., 2015. Department of Mathematics, University of Wisconsin, Madison, WI 53706 E-mail address:
[email protected] URL: http://www.math.wisc.edu/~ellenber/ Delft Institute of Applied Mathematics, Delft University of Technology E-mail address:
[email protected] URL: http://homepage.tudelft.nl/64a8q/
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